On J-Colorability of Certain Derived Graph Classes

Federico Fornasiero

Department of mathemathic Universidade Federal de Pernambuco Recife, Pernambuco, Brasil [email protected]

Sudev Naduvath

Centre for Studies in Discrete Mathematics Vidya Academy of Science & Technology Thrissur, Kerala, India. [email protected]

Abstract

A vertex v of a given graph G is said to be in a rainbow neighbourhood of G, with respect to a proper coloring C of G, if the closed neighbourhood N[v] of the vertex v consists of at least one vertex from every colour class of G with respect to C. A maximal proper colouring of a graph G is a J-colouring of G if and only if every vertex of G belongs to a rainbow neighbourhood of G. In this paper, we study certain parameters related to J-colouring of certain Mycielski type graphs.

Key Words: Mycielski graphs, graph coloring, rainbow neighbourhoods, J-coloring

arXiv:1708.09798v2 [math.GM] 4 Sep 2017 of graphs.

Mathematics Subject Classiﬁcation 2010: 05C15, 05C38, 05C75.

1 Introduction

For general notations and concepts in graphs and digraphs we refer to [1, 3, 13]. For further deﬁnitions in the theory of graph colouring, see [2, 4]. Unless speciﬁed otherwise, all graphs mentioned in this paper are simple, connected and undirected graphs.

1 2 On J-colorability of certain derived graph classes

1.1 Mycielskian of Graphs

Let G be a triangle-free graph with the vertex set V (G) = {v1, . . . , vn}. The Myciel- ski graph or the Mycielskian of a graph G, denoted by µ(G), is the graph with vertex set V (µ(G)) = {v1, v2, . . . , vn, u1, u2, . . . , un, w} such that vivj ∈ E(µ(G)) ⇐⇒ vivj ∈ E(G), viuj ∈ E(µ(G)) ⇐⇒ vivj ∈ E(G) and uiw ∈ E(µ(G)) for all i = 1, . . . , n.

u1 u2 u3 u4 u5 u6 u7

v1 v2 v3 v4 v5 v6 v7

Figure 1: The Mycielski graph µ(P7)

In the above mentioned conditions of Mycielski graphs, we call the two vertices vi, ui twin vertices and the vertex w is called the root vertex of the Mycielskian µ(G). By a Mycielski type graph, we mean a graph that can be constructed from the Mycielski graphs or the graphs generated from a given graphs using some or similar rules of constructing their Mycielski graphs.

1.2 J-Coloring of Graphs The notion of J-coloring of a graph, has been deﬁned for the ﬁrst time in [10] as follows:

Deﬁnition 1.1. [10] A graph G is said to have a J-colouring C if it has the maximal number of colours such that if every vertex v of G belongs to a rainbow neighbourhood of G. The number of colours in a J-coloring C of G is called the J-colouring number of G.

Deﬁnition 1.2. [10] A graph G is said to have a J ∗-colouring C if it has the maximal number of colours such that if every internal vertex v of G belongs to a rainbow neighbourhood of G. The number of colours in a J ∗-coloring C of G is called the J ∗-colouring number of G.

It can be noted that all graphs, in general, need not have a J-coloring. Hence, the studies on the graphs which admit J-coloring and their properties and structural characterisations attract much interests. Some studies in this direction can be seen in [6, 10]. The initial purpose of this paper is to study the J-colourability of the Myciel- skian and certain Mycielski type graphs of some fundamental graph classes. Federico Fornasiero and Sudev Naduvath 3

2 J-Colourability of Mycielski Graphs

Note that the Mycielski graph µ(G) of a graph G has no pendant vertices and hence the J-colouring and the J ∗-colouring of the Mycielski graphs µ(G), if they exist, are the same. We ﬁrst try to repeat the original demonstration of Mycielskian graph. To do that, we have to ﬁx a J-colouring on the graph G.

Theorem 2.1. Let G be a graph with J-colouring C = {c1, c2, c3 . . . , ck}. Then, the graph µ(G) is not J-colourable (and so, it is not J ∗-colourable).

Proof. Note that µ(K2) is isomorphic to C5 and hence it is not J −colourable (as it is proved in [10]). Hence, we can consider graphs with order greater than 2. Let us assume that we can have a new J-colouring C = {c1, c2 . . . , cj} on µ(G). Without 0 loss of generality, we can assume now that the colour of w is c1. Every vertex ui have to be coloured with one of the other colour c2, . . . , cj. Hence, there exists at least a vertex v1 with colour c1. Let be v2 the vertex connected to v1 such as the twin vertex u2 has the colour c2. Here, we have the following two possibilities:

(i) : If the colour of v2 is diﬀerent from the previous ones, let us say c3, we have that for the deﬁnition of rainbow neighbourhood even the twin vertex u2 has to be connected with a vertex with the same colour, and it has to be a vertex v3 because no one of the vertices uj are connected, and w has the colour c1. But if it is so, than for the construction of µ(G) even the vertex v2 has to be connected with v3, and so it is not a proper coloring because two vertices has the same colours (see Figure 2).

w c1

u1 c2 c2 u2

v3 c3 v1 c1 c3 v2

Figure 2: Case (i)

(ii) If the colour of v2 is c2, the twin vertices u1 of v1 has to have a different colour 0 (let us say c3), because it is linked to w that has the colour c1 and to v2 that has the colour c2. But, in this case the vertex v1 has to be connected with another vertex which has the colour c3. So we have to differentiate two different possibilities:

(ii)(a) If the vertex v1 is connected to a vertex v3 who has the colour c3 for the construction of µ(G) even the twin vertices u1 is connected to v3 and so it 4 On J-colorability of certain derived graph classes

is not a proper coloring because two connected vertices have the same colour (see Figure 3).

w c1

c3 c2 u1 u2

c3 c1 c2 v3 v1 v2

Figure 3: Case (ii)(a)

(ii)(b) If the vertex v1 is connected to a vertex u3 which has the colour c3, ﬁrst we can note that the twin vertex v3 cannot have a new colour, because it would lead to a contradiction over u3 similar to the case (i).

Note that v3 cannot have the colour c1 (because for construction it is connected with v1) nor the colour c3 (because always for construction it is connected to the twin vertex u1) so it has to be coloured with the colour c2. But if it is so, u3 needs to be connected with a vertex vi whose colour is c2, but it cannot be the twin vertex v3 for the construction, nor the vertex v2 because it will lead to have the triangle v1v2u3v1. If the graph G has only 3 nodes we reach a contradiction yet, if it is not let us call v4 the vertex with colour c2 connected to u3. For the construction of µ(G) it has to be connected to the vertex v3 and it ﬁnally leads to a contradiction because the two vertices would have the same colour (see Figure 4).

w c1

c2 c3 c3 u2 u1 u3

c2 c1 c2 c2 v2 v1 v3 v4

Figure 4: Case (ii)(b)

Hence, the Mycielskian graph of any graph G is not J-colourable, irrespective of whether the G is J-colourable or not. Federico Fornasiero and Sudev Naduvath 5

3 Some New Constructions

Since Mycielskian of any graph does not admit a J-colouring, our immediate aim is to construct some simple connected graphs from certain given graphs such that new graphs also admit an extended J-coloring. In this section, we discuss the J-colorability of certain newly constructed Mycielski type graphs of a given graphs. The ﬁrst one among such graphs is the crib graph, denoted by c(G), of a graph G, which is deﬁned in [11] as follows:

Deﬁnition 3.1. [11]The crib graph, denoted by c(G), of a graph G is the graph whose vertex set is V (µ(G)) = {v1, v2, . . . , vn, u1, u2, . . . , un, w} such that vivj ∈ E(µ(G)) ⇐⇒ vivj ∈ E(G), viuj ∈ E(µ(G)) ⇐⇒ vivj ∈ E(G) and viw, uiw ∈ E(µ(G)) for all i = 1, . . . , n.

Figure 5 depicts the crib graph of P6.

u1 u2 u5 u6 u3 u4

v1 v2 v3 v4 v5 v6

Figure 5: Crib graph of P6

The following theorem discusses the admissibility of an extended J-coloring by the crib graph of a J-colorable graph G.

Theorem 3.2. The crib graph c(G) of a J-colourable graph G is also J-colourable. Also, J(c(G)) = J(G) + 1.

Proof. Assume that the graph G under consideration admits a J-coloring, say C = {c1, c2, . . . , ck}, where k = χ(G), the chromatic number of G. While coloring the vertices of c(G), we notice the following points:

(i) Since, the twin vertices ui and vi in c(G) are adjacent to each other, both of them can have the same color.

(ii) Since N(ui) = N(vi) for all 1 ≤ i ≤ n, it follows that N[ui] is also a rainbow neighbourhood in c(G). Therefore, the subgraph of c(G) induced by the vertex set {v1, v2, . . . , vn, u1, u2, . . . , un} admit the same J-coloring C. (iii) Since the root vertex w is adjacent to other vertices in c(G), it cannot have any color from C. Therefore, we need a new color, say ck+1 to color the vertex w. 6 On J-colorability of certain derived graph classes

(iv) Since the root vertex w is adjacent to other vertices in c(G), it belongs to a rainbow neighbourhood in c(G) and will not inﬂuence the belongingness of other vertices to some rainbow neighbourhoods in c(G).

In view of the conditions mentioned above, notice that C ∪{ck+1} is a J-coloring of c(G) and J(c(G)) = k + 1 = J(G) + 1. This completes the proof. Another similar graph that catches attention in this context is the shadow graph of a graph G. The shadow graph of a graph G, denoted by s(G), is the graph G is the graph obtained from its Mycielski graph µ(G) by removing the root vertex. The following theorem discusses the admissibility of a J-coloring by the shadow graph s(G) of a J-colorable graph G.

Theorem 3.3. The shadow graph s(G) of a J-colorable graph G is also J-colorable. Moreover, J(s(G)) = J(G).

Proof. The proof is immediate from the proof of Theorem 3.2. Next, we construct a new graph F(G) from a triangle-free, simple and connected graph G such that F(G) has J-chromatic number k + 1 when G has J-chromatic number k. The construction is described below.

Deﬁnition 3.4. Let G be a triangle-fee graph, with V (G) = {v1, . . . , vn}. We deﬁne the Federico graph F(G) of G as the graph such that V (F(G)) = {v1, v2 . . . , vn, u1, u2 ..., un, w1, w2 . . . , wn} and with edges that follows the rules:

(i) vivj ∈ E(F(G)) ⇐⇒ vivj ∈ E(G)

(ii) wiwj ∈ E(F(G)) ⇐⇒ vivj ∈ E(G)

(iii) uiwj ∈ E(F(G)) ⇐⇒ vivj ∈ E(G)

(iv) for all i = 1, . . . , n, viui ∈ V (F(G))

The following ﬁgure illustrates the Federico graph of the graph P5.

w1 w2 w3 w4 w5

u1 u2 u3 u4 u5

v1 v2 v3 v4 v5

Figure 6: The Federico Graph F(P5)

First we can note that the graph F(G) has no pendant vertices and so the J- colouring od F(G), if exists, coincides to the J ∗-colouring. This fact is straight forward. Federico Fornasiero and Sudev Naduvath 7

Theorem 3.5. Let G be a J-colourable, triangle-free graph of order n with J- colouring number k. Then the graph F(G) is triangle-free and with higher J- colouring number. If J(G) = k, then J(F(G)) = k + 1.

Proof. First of all we can see that no pair of vertices ui is connected, therefore no triangle can involve a pair of these vertices. Also, no vertex wi is connected to a vertex vj. Remembering that G is triangle-free, it is not possible that three vertices vi are connected in F(G) too. Similarly for the vertices wi that form between them a copy of the graph G. Hence, we have only two possibilities left:

(i) if vi is connected to vj we have that ui is connected to vi but not to vj, by construction, so no triangle of this type is involved.

(ii) if wi is connected to wj we have that ui is connected to wj but not to wi, so it is proved that F(G) is triangle-free. To construct a proper J-colouring on F(G), let consider a proper J-colouring ∗ ϕ : V (G) → {c1, . . . , ck} and let us construct ϕ : V (F(G)) → {c1, . . . , ck, ck+1} by setting:

∗ ∗ (i) ϕ (vi) = f (wi) = f(vi) for all i = 1, . . . , n ∗ (ii) ϕ (ui) = ck+1 for all i = 1, . . . , n First, we have to prove that it is a proper J-colouring of F(G). We note that every vertex vi has a rainbow neighbourhood in G, and hence it has in µ(G) with one more colour (the colour of ui). Every wi has the same rainbow of the twin vi because it is connected with the same vertices connected to vi, and it is connected at least to one of the vertex uj. Finally, every ui has a k +1 rainbow neighbourhood of because it is connected with every vi and with every wj that are the connections ∗ of vi in the original graph, so by the definition of ϕ , every ui has the same rainbow neighbourhood of vi. Hence, this colouring define a proper J-colouring of G, it remains to prove that this colouring is maximal. Hence, let us assume that there exists a proper J-colouring of F(G) such that J(F(G)) = 2. In this case, we can assume that not every ui has the same colour because if not every ui is connected only to vi, and every vi has a rainbow neighbourhood of order k + 2 and can’t have the same colour of the vertices ui. But it would mean that the graph G was (k + 1) − J−colouring. Hence, let us start considering the vertex ui. If we prove that independent from the choice of the colour of ui, it is necessary that every ui has the same colour, for what we have just proved, it follows that the coloring is maximal. From the above choice of the coloring assignment ϕ∗, we note that F(G) requires at least one more color in its proper coloring than the corresponding proper coloring of the graph G. Now, note that the upper bound for the J-chromatic number of a graph G is δ(G)+1 (see [10]). Since δ(F(G)) = δ(G)+1, any J-coloring of F(G) can have at most one more color than the J-coloring of G. From these two conditions, we can conclude that the coloring ϕ∗ defined above is a maximal coloring of F(G) such that every vertex of F(G) belongs to some rainbow neighbourhood of F(G). Then, we have J(F(G)) = J(G) + 1, completing the proof. 8 On J-colorability of certain derived graph classes

Hence, we have found an interesting construction to have new triangle free graphs with higher J-colouring number. In the following theorem, we study what happens to the chromatic number of a Federico graph.

Theorem 3.6. Let G be a graph and F(G) its Federico graph. Then, χ(G) = χ(F(G))

Proof. Let f : V (G) = {v1, . . . , vn} → c1, . . . , ck be a coloring of the vertices of G. Let us consider the coloring g : V (F(G)) = {v1, . . . , vn, u1, . . . , un, w1, . . . , wn} → {c1, . . . , ck} deﬁned by:

(i) g(ui) = g(wi) = f(vi) for all i = 1, . . . , n.

(ii) if f(vi) = ch then g(vi) = ch+1 for all i = 1, . . . , n and h = 1, . . . , k − 1

(iii) if f(vi) = ck then g(vi) = c1 for all i = 1, . . . , n.

To prove that it is a proper coloring, first we can note that it is a proper coloring on the vertices vi because f was a proper coloring of G and we have only permutated the colours, and also it is a proper coloring on the vertices wi because it is a copy of the graph and we have coloured in the same way. So it only left to see that we cannot have the same colour with connections with a vertex ui. But, vi is only connected to ui and they have different colours because g(ui) = f(vi) but ch+1 = g(vi) 6= f(vi) = ch for the definition of g. Also, because each wi is connected to every uj such that vj ∈ V (G) was connected to vi ∈ V (G) and none of which has the same colour g(ui) = f(ui) no conflicts arise here. Hence, we have constructed a proper coloring of F(G) with the same number of colours of G, as claimed. Now we want to study another important colouring property of the Modified Mycielski graph, the circular chromatic number. It was first studied in [12] with the name of star chromatic number, and later in [14] provided a comprehensive survey. Let G be a graph. For two positive numbers k, d with k ≥ 2d, we define a (k, d)- colouring as the function f : V (G) → {0, 1, . . . , k − 1} such that if two vertices u, v are adjacent, then |f(u) − f(v)|k ≤ d where |a − b|k = min{|a − b|, k − |a − b|}. Then, the circular chromatic number of G is defined as

k χ (G) := inf | G has a (k, d) − coloring c d

In [14] it is shown that if the graph G has at least one edge, then the inﬁmum can be replaced with the minimum and we have χ(G) − 1 ≤ χc(G) ≤ χ(G). The circular chromatic number is hard to compute in Mycielski graphs and there’s not yet a general formula that compute χc(µ(G)) knowing the circular chromatic number of G. But, in the case of Federico graph, we have

Theorem 3.7. Let G be a graph with χc(G). Then, χc(F(G)) = χc(G).

Proof. Let G be a graph with a (k, d) coloring f over V (G) = {v1, v2 . . . , vn}. Then, we construct the coloring f ∗ over F(G) as follows: Federico Fornasiero and Sudev Naduvath 9

∗ (i) f (vi) = f(vi) for all i = 1, . . . , n ∗ ∗ (ii) f (ui) = f (wi) = f(vi) − d mod k for all i = 1, . . . , n To see that it’s a proper (k, d) coloring of F(G) we ﬁrst note that between it’s a proper coloring over the vertices vi’s (because it was on G) and over the vertices wi’s (because in the construction we have simply added the distance modulo k, and their connections are the same than the connections over the vertices vi). A vertex vi is adjacent only to the vertex ui and so by construction it has exactly distance d. The vertex ui is connected to every vertex wj such that vivj is an edge in G. But for construction the vertex ui has colour f(vi) − d mod k and the vertices wj have colour f(vj) − d mod k, so the connection maintain the same distances over k the edges vivj ∈ E(G). Hence, it is a proper (k, d)−colouring of F(G) and if d is minimal in G so it’s minimal in F(G).

4 J-Paucity Number of Graphs

In view of our results on the absence of J-coloring for Mycielski graphs and our new constructions from the Mycielski graphs which admit J-colorings, we define a new graph parameter with respect to J-coloring as follows: Definition 4.1. Let G be a graph which does not admit a J-coloring. Then, the J-paucity number of G, denoted by %(G), is defined as the minimum number of edges to be added to G so that the reduced graph becomes J-colorable with respect to a (δ(G) + 1)-coloring of G. In the following theorem, we determine the J-paucity number of paths.

Theorem 4.2. %(µ(Pn)) = n.

Proof. Note that for δ(µ(Pn)) = 2 and hence we have to find the minimum number of edges to be added to µ(Pn) so that the reduced graph becomes J-colorable using 3 colors. For this, first assign colors c1 and c2 alternatively to the vertices v1, v2, . . . , vn. Now color the vertices ui such that ui and its twin vertex vi have the same color. Since the vertex w is adjacent to all ui’s, it can be seen that it must have a different color, say c3 (see Figure 7). w c3

c1 c2 c1 c2 c1 c2 c1 u1 u2 u3 u4 u5 u6 u7

c1 c2 c1 c2 c1 c2 c1 v1 v2 v3 v4 v5 v6 v7

Figure 7: A 3-coloring of µ(P7).

We notice the following points in this context: 10 On J-colorability of certain derived graph classes

(i) No vertex vi in V (µ(Pn)) belongs to a rainbow neighbourhood of µ(Pn), as none of them is adjacent to a vertex having color c3;

(ii) Every vertex ui with color c2 is adjacent to at least one vertex vj with color c2 and the vertex w with color c3, thus belonging to some rainbow neighbourhood in µ(Pn).

(iii) Every vertex uj with color c2 is adjacent to at least one vertex vk with color c1 the vertex w with color c3, thus belonging to some rainbow neighbourhood in µ(Pn).

(iv) The vertex w, being adjacent to all vertices ui, belongs to some rainbow neighbourhoods in µ(Pn).

Therefore, from the above arguments, what we need is to draw edges from the vertices vi to the vertex w so that they also are in some rainbow neighbourhoods of G. Therefore, %(µ(Pn)) = n.

Theorem 4.3. %(µ(Cn)) = n + 2r, where r ∈ N is given by n ≡ r(mod 3).

Proof. Since δ(µ(Cn)) = 3, the maximum number of colors in its J-coloring is 4. Hence, we have to ﬁnd the minimum number of edges to be added to µ(Pn) so that the reduced graph becomes J-colorable using 4 colors. Here we have to consider the following cases: Case-1 : Let n ≡ 0(mod 3). Then, we can assign colors c1, c2 and c3 alternatively to the vertices v1, v2, . . . , vn. As mentioned in the previous result, we can color the vertices ui such that ui and its twin vertex vi have the same color. Since the vertex w is adjacent to all ui’s, it must have a diﬀerent color, say c4 (see Figure 8). In this case, all vertices ui and the vertex w will belong to some rainbow neighbourhoods of µ(Cn), but no vertex vi has an adjacent vertex having colour c4. So, we need to draw edges from all vi; 1 ≤ i ≤ n to the vertex w in order to include them in some rainbow neighbourhoods of µ(Cn). Case-2 : Let n ≡ 1(mod 3). Then, we can assign colors c1, c2 and c3 alternatively to the vertices v1, v2, . . . , vn−1. The vertex vn can be colored only by c2, as it is adjacent to v1 with color c1 and to vn−1 with color c3. Here, we notice that the vertex v1 is not adjacent to any vertex having color c3. Here, we need to draw an edge between v1 and one of the vertices having color c3. If we label the vertices ui in such a way that the twin vertices have the same color, then as in the case of v1, the vertex u1 is not adjacent to any vertex of color c3. Hence, we need to draw an edge from u1 to any one of the vertices having color c3. Since w is adjacent to all ui, w must have the fourth color c4. Since every vertex ui is adjacent to w, all these vertices,(except u1) belong to some rainbow neighbourhood of µ(Cn). (Also, note that when we draw an edge from u1 to a vertex having color c3, it will also belong to some rainbow neighbourhood). Since no vertex vi is adjacent to a vertex having colour c4, each of them is to be connected to the vertex w by a new edge. Therefore, in this case %(µ(Cn)) = n + 2. Case-3 : Let n ≡ 2(mod 3). Then, we can assign colors c1, c2 and c3 alternatively to the vertices v1, v2, . . . , vn−2. Then, the vertex vn−1 gets the color c1, the vertex Federico Fornasiero and Sudev Naduvath 11

v8 v7 c2 c1

c3 v9 u8 u7 c2 c1 v6 c3 c3 u9 u6 c3

w c4 c1 u1 c1 v1

u5 c2 c2 u2 v5 c2 c1 c3 u4 u3 c2 v2

c1 v4 c3 v3

Figure 8: A minimal proper coloring of µ(C9)

v1 can have the color color c2 Note that the vertex v1 and vn are not adjacent to any vertex having color c3. Here, we need to draw one edge each from v1 and v2 to some vertices having color c3. If we label the vertices ui in such a way that the twin vertices have the same color, then as in the case of v1 and v2, the vertices u1 and un will not be adjacent to any vertex of color c3. Hence, we need to draw one edge each from u1 and u3 to some of the vertices having color c3. The vertex w gets the color c4 and as mentioned in the above cases, we need to draw edges from all vertices vi to w so that all vertices in µ(Cn) belong to some rainbow neighbourhoods in µ(Cn). Therefore, in this case %(µ(Cn)) = n + 4.

5 Conclusion

In this paper, we have proved that the Mycielskian of any graph G will not have a J-coloring, irrespective of whether G has a J-coloring or not. We have also checked the existence of J-coloring for certain new Mycielski type graphs constructed from certain graphs. There is a wide scope for further studies in this area by exploring for new and related graph constructions. We have also investigated the possibility of deﬁning J-colorings for given graphs by adding new edges between their non-adjacent vertices. Furthermore, we have determined the minimum number of such edges to be introduced for the Mycielskian of paths and cycles. The studies in this area for more graph classes and more derived graphs are also promising. 12 On J-colorability of certain derived graph classes

Declaration

The authors declare that they have no competing interests. Both the authors con- tributed signiﬁcantly in writing this article. The authors read and approved the ﬁnal manuscript.

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